WORKING PAPER

Synthetic Control Estimation Beyond Case Studies Does the Minimum Wage Reduce Employment?

David Powell

RAND Labor & Population

WR-1142 February 2016 This paper series made possible by the NIA funded RAND Center for the Study of Aging (P30AG012815) and the NICHD funded RAND Population Research Center (R24HD050906).

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Synthetic Control Estimation Beyond Case Studies: Does the Minimum Wage Reduce Employment?∗ David Powell RAND † First Version: October 2015 This Version: February 2016 Abstract Panel data are often used in empirical work to account for additive fixed time and unit effects. More recently, the synthetic control estimator relaxes the assumption of additive fixed effects for case studies, using pre-treatment outcomes to create a weighted average of other units which best approximate the treated unit. The synthetic control estimator is currently limited to case studies in which the treatment variable can be represented by a single indicator variable. Applying this estimator more generally, such as applications with multiple treatment variables or a continuous treatment variable, is problematic. This paper generalizes the case study synthetic control estimator to permit estimation of the effect of multiple treatment variables, which can be discrete or continuous. The estimator jointly estimates the impact of the treatment variables and creates a synthetic control for each unit. Additive fixed effect models are a special case of this estimator. Because the number of units in panel data and synthetic control applications is often small, I discuss an inference procedure for fixed N . The estimation technique generates correlations across clusters so the inference procedure will also account for this dependence. Simulations show that the estimator works well even when additive fixed effect models do not. I estimate the impact of the minimum wage on the employment rate of teenagers. I estimate an elasticity of -0.44, substantially larger than estimates generated using additive fixed effect models, and reject the null hypothesis that there is no effect. Keywords: synthetic control estimation, finite inference, minimum wage, teen employment, panel data, interactive fixed effects, correlated clusters JEL classification: C33, J23, J31

∗

I thank both the Center for Causal Inference and the Methods Lab for providing funding. I am also appreciative of both Arindrajit Dube and David Neumark for making their respective data and code available. I received helpful feedback from seminar participants at the Center for Causal Inference as well as Abby Alpert, Tom Chang, John Donohue, Michael Dworsky, Beth Ann Griffin, Kathleen Mullen, David Neumark, and Michael Robbins. † Contact information: [email protected]

1

1

Introduction

Empirical research often relies on panel data to account for fixed differences across units and common time effects, using changes in the policy variables to identify the relationship between the policies and the outcome of interest. Additive fixed effect models include separate additive unit (“state”) fixed effects and additive time fixed effects. These models assume that an unweighted average of all states acts as a valid counterfactual for each state. In this paper, I introduce a synthetic control estimator which creates an empirically-optimal control group for each unit using a weighted average of the other units. This estimator builds on and generalizes previous work by Abadie and Gardeazabal (2003), Abadie et al. (2010), and Abadie et al. (2014). This previous work introduced a technique to estimate causal effects for case studies and I refer to the estimator as the “case study synthetic control” (CSSC) estimator. The innovation of this approach was to use the period before the treatment (terrorism outbreak in Abadie and Gardeazabal (2003); the adoption of a tobacco control program in California in Abadie et al. (2010)) to empirically construct a comparison group which is appropriate for the group exposed to the treatment. The synthetic control estimator uses a data-driven approach to create this comparison group. A typical motivation for the use of the synthetic control estimator is that the parallel trends assumption required by additive fixed effect models often does not hold in practice. It has become common to cite the “parallel trends” assumption implicit in additive fixed effect models and to test for pre-existing trends through event study analysis. If the event study analysis suggests non-parallel trends, however, it is not generally clear how to proceed. CSSC permits estimation of specifications with interactive fixed effects which generate non-parallel trends. The estimator introduced in this paper generalizes CSSC beyond case studies, per2

mitting the inclusion of multiple discrete or continuous treatment variables in the main specification as well as interactive fixed effects. The estimator jointly estimates the parameters associated with the treatment variables and a synthetic control for each unit. The estimator performs well in simulations. I apply this estimator to study the employment effects of the minimum wage. Understanding the employment ramifications of minimum wage increases is an essential part of the debate over the efficacy of federal, state, and local policies to change the minimum wage. The minimum wage rose in 14 states at the beginning of 2016 (12 due to legislative action and 2 due to automatic cost-of-living increases).1 President Barack Obama promoted a bill to raise the federal minimum wage to $10.10 in his 2014 State of the Union Address and reiterated his support for a minimum wage increase in his 2015 and 2016 State of the Union Addresses. Presidential candidate Bernie Sanders supports a federal minimum wage increase to $15 per hour while candidate Hillary Clinton supports an increase to $12 per hour. The economic literature on the relationship between the minimum wage and employment has exploited state-level minimum wage increases to test whether states with increasing minimum wage experience different employment changes relative to states with stagnant minimum wages over the same time period. This additive fixed effects approach assumes that adopting states and non-adopting states would have experienced the same employment trends, on average, in the absence of the policy adoption. Recent work has questioned the usefulness of this approach with evidence that estimated disemployment effects in the traditional fixed effects approach are not robust to the inclusion of state-specific trends (Allegretto et al. (2011)) or limiting comparisons to contiguous counties which straddle state borders (Dube et al. (2010)). In response, Neumark et 1

http://www.ncsl.org/research/labor-and-employment/state-minimum-wage-chart.aspx

3

al. (2014b) argues that these new approaches do not produce more appropriate comparison groups and estimates a disemployment elasticity for teenagers of -0.15. In more recent work, Neumark et al. (2014a) states, “A central issue in estimating the employment effects of minimum wages is the appropriate comparison group for states (or other regions) that adopt or increase the minimum wage.” Synthetic control estimation offers a method to empirically select appropriate control groups for treated states. The application of CSSC has proven quite useful in applied work (see Nonnemaker et al. (2011); Hinrichs (2012); Cavallo et al. (2013); Saunders et al. (2014); Ando (2015); Bauhoff (2014); Fletcher et al. (2014); Cunningham and Shah (2014); Mideksa (2013); Torsvik and Vaage (2014); Munasib and Rickman (2015); Munasib and Guettabi (2013); Eren and Ozbeklik (2015) for a small subset of examples). While initially motivated by case study applications, the CSSC estimator is often used in circumstances where multiple units are treated (e.g., Billmeier and Nannicini (2013); Donohue et al. (2015); Powell et al. (2015)) by applying the CSSC estimator to each treated unit and then aggregating the estimates, though there is little agreement on how to implement this aggregation. The treatment in these circumstances can be represented by a single indicator variable. In this paper, I extend the synthetic control approach beyond case studies and applications where the treatment variable must be represented by a single indicator variable. The synthetic control estimator introduced in this paper allows for multiple treatment variables, which can be discrete or continuous. I will refer to this estimator as the generalized synthetic control (GSC) estimator. This estimator parallels fixed effects estimators which account for fixed state differences and time fixed effects while allowing for multiple discrete or continuous treatment variables. However, instead of assuming additive state fixed effects and additive year fixed effects, the synthetic control estimator permits estimation of a specification with interactive fixed effects while nesting more traditional additive fixed effect models. Additive

4

fixed effect models are equivalent to subtracting off the mean value of the outcome and the explanatory variables for each year, giving equal weight to each state for this de-meaning. The synthetic control estimator generates a data-driven control for each state which does not constrain the weights on the units to be equal, assigning different weights to generate an optimal control. A primary motivation of CSSC in many applications is that it allows for the creation of a synthetic control which has a similar outcome trajectory in the pre-treatment period, typically implemented by using the outcome in each pre-treatment period to create the synthetic control unit. Kaul et al. (2015) points out that if every pre-treatment outcome is used for the creation of the synthetic control, then additional covariates are not used by CSSC. GSC, by allowing for multiple explanatory variables, circumvents this problem and uses the control variables in a way that is similar to linear panel data models – by estimating the independent relationship between each control variable and the outcome. This is a further advantage of using GSC, even with case studies. Similarly, in the circumstance of multiple treated units of the same policy, GSC provides a natural means of aggregating the estimates across adopting units. For inference, Abadie and Gardeazabal (2003) suggests a placebo test. Recent work has questioned the validity of this method for CSSC inference (Ando and S¨avje (2013)) and it is difficult to apply placebo tests more generally when the treatment is continuous or there are multiple treatment variables. I build on a method developed for panel data in complementary work (Powell (2015)) which provides valid p-values in the presence of a finite number of correlated clusters. Accounting for dependence across units and developing appropriate inference for fixed N are crucial for use with a synthetic control estimator. First, many panel data applications involve only a relatively small number of units. Second, the estimator uses other states as controls for each unit, mechanically inducing correlations

5

across units. In the next section, I further discuss the CSSC estimator to motivate a more general synthetic control technique and include background on the minimum wage debate in the economics literature. In Section 3, I introduce the generalized synthetic control (GSC) estimator and discuss an appropriate inference procedure. Section 4 includes simulation results. In Section 5, I apply GSC to estimate the relationship between state minimum wages and the employment rate of teenagers. Section 6 concludes.

2 2.1

Background Case Study Synthetic Control Estimation

This paper builds on the synthetic control estimation technique discussed in Abadie and Gardeazabal (2003), Abadie et al. (2010), and Abadie et al. (2014). In the CSSC framework, there are J + 1 units and T time periods. Unit 1 is exposed to the treatment in periods T0 + 1 to T and unexposed in periods 1 to T0 . All other units are unexposed in all time periods. Outcomes are defined by a factor model:

Yit = αit Dit + δt + λt μi + it

(1)

where Dit represents the treatment variable and is equal to 1 for state i = 1 and time periods t > T0 , 0 otherwise. λt is a 1 × F vector of common unobserved factors and μi (F × 1) is a vector of factor loadings. Gobillon and Magnac (forthcoming) compares the synthetic control approach to estimation using interactive fixed effects (Bai (2009)) for estimation of equation (1). This work illustrates the usefulness of relaxing the restriction that the fixed effects are additive. 6

Unit 1 is the treated unit while all other units are part of the “donor pool,” the units that may potentially compose part of the synthetic control. The motivation for the approach is to find a weighted combination of units in the donor pool such that J+1 

wj∗ μj = μ1 .

(2)

j=2

If this condition holds, then

J+1 j=2

wj∗ Yjt provides an unbiased estimate of Y1t for D1t =

0. The weights are constrained to be non-negative and to sum to one. They are generated to minimize the difference between the pre-treated outcomes of the treated unit and the unit’s synthetic control. I assume that there are no other control variables, which are denoted Zit in Abadie and Gardeazabal (2003) and included additively in equation (1), and that all pre-treatment outcomes (and only pre-treatment outcomes) are used to generate the synthetic control weights. This is common in applications using CSSC (see Cavallo et al. (2013) for one example).2 The inclusion of control variables to generate the match between the treated unit and the synthetic control has been criticized (Kaul et al. (2015)) as the CSSC estimator will not use the covariates when each value of the pre-treatment outcome is used. For this reason, I assume that no additional covariates are used with the CSSC estimator. The GSC estimator below will allow for the use of control variables in a manner that parallels traditional regression analysis by letting them have a direct effect on the outcome variable and estimating that effect jointly with the other parameters. Let W = (w2 , . . . , wJ+1 ) such that wj ≥ 0 for all 2 ≤ j ≤ T + 1 and

J+1 j=2

wj = 1.

Let X1 = (Y11 , . . . , Y1T0 ), a (T0 × 1) vector of pre-intervention outcomes for the treated unit. X0 is a (T0 × J) of the same variables for the units in the donor pool. The weights are 2

The use of each pre-treatment outcome to create the synthetic control is especially useful to account for differential pre-existing (non-linear) trends across units.

7

estimated using   W∗ = argmin |X1 − X0 W|V W

s.t. wj ≥ 0 for all 2 ≤ j ≤ T + 1 and

J+1 

wj = 1

j=2

for some matrix V. In words, CSSC involves a constrained optimization to find the weighted average of the donor states which is “closest” to the treated unit in terms of pre-treated outcomes. The treatment effect estimate for period t > T0 is

α ˆ it = Yit −

J+1 

wj∗ Yjt .

(3)

j=2

CSSC permits estimation of a treatment effect for each post-treatment time period, though it is common in applied work to report an aggregate post-treatment effect. For inference, Abadie and Gardeazabal (2003) and Abadie et al. (2010) recommend a placebo test in which a treatment effect is estimated for each unit in the donor pool as if it were treated (and Unit 1 is included in the donor pool). Under the null hypothesis that there is no treatment effect, these placebo tests generate a distribution of estimates which would be observed randomly. The placement of α ˆ it in this distribution generates an appropriate p-value. Ando and S¨avje (2013) discusses possible limitations of the placebo test approach for synthetic control estimation. CSSC has proven valuable in applied work as a means of relaxing the parallel trends assumption required by additive fixed effect models. The framework of equation (1) nests

8

the more typical additive fixed effect model which assumes  λt =

 1 φt

⎡

⎤

⎢ γi ⎥ and μi = ⎣ ⎦, 1

(4)

such that λt μi = γi +φt . When additive fixed effects are appropriate, CSSC will, in principle, apply appropriate weights. The application of CSSC has proven quite useful in applied work. A primary contribution of this paper is that the motivation behind CSSC should apply more generally to panel data models, not just case studies.

2.2

Finite Inference

Panel data models are often applied in cases where the number of units is relatively small. For example, the minimum wage analysis of this paper uses 50 states plus Washington D.C. The donor pool in Abadie et al. (2010) includes 38 states; Abadie and Gardeazabal (2003) uses 16 Spanish regions; Abadie et al. (2014) uses a sample of 16 OECD countries. One exception in the CSSC context is Robbins et al. (2015) which studies a crime intervention policy implemented by city block and includes a total of 3,601 blocks in the data. Similarly, Ando (2015) has over 800 units in the donor pool. In this paper, I use an inference procedure for fixed N , which is useful more generally for panel data with a fixed number of units. This method is introduced in Powell (2015) and applied here for the synthetic control estimator as a special case. Additive fixed effects create problems when N is small because they induce a mechanical correlation across clusters. Inference procedures developed for a small number of clusters are frequently not valid for estimation with state and year fixed effects (e.g., Hansen (2007); Bester et al. (2011); Ibragimov and M¨ uller (2010)). The synthetic control estimator introduced in this paper also

9

generates a correlation across clusters as the synthetic control for each state is a weighted average of other states. After subtracting off each state’s synthetic control, the gradient of each unit is a function of the error term for that unit and a combination of other units. I develop an inference procedure which accounts for empirical correlations across units. These correlations can also arise for reasons other than the estimation procedure, such as similarities across states due to geography, political institutions, industry composition, etc.

2.3

Minimum Wage and Employment

A vast literature has debated whether minimum wage increases affect employment rates (see Neumark et al. (2014b) for a review). The empirical literature has recently focused on determining appropriate controls for states with increasing minimum wages. State minimum wages increase due to state legislative changes, binding federal minimum wage changes, or state cost-of-living adjustments. The literature typically uses all sources of variation, comparing states with increasing minimum wages to states without a change in their minimum wage. The potential of the CSSC estimator to generate appropriate control units has been recognized recently in the literature, though it it difficult to appropriately apply CSSC to this application. I discuss uses of the CSSC estimator to study the possible disemployment effects of the minimum wage in the literature. Sabia et al. (2012) studies the employment effects of the 2004-2006 New York minimum wage increase, finding large employment reductions. They use CSSC for part of their analysis, though they do not use any pre-intervention outcomes to construct the synthetic control. Instead, the synthetic control is based only on other covariates which is unlikely to account for pre-existing trends. More recently, Dube and Zipperer (2015) (also discussed in Allegretto et al. (2015))

10

uses CSSC to study the minimum wage employment effects, discussing the application of CSSC for a continuous treatment variable. To implement CSSC, the authors select on states with no minimum wage changes for two years prior to the treatment (i.e., minimum wage change) and with at least one year of post-treatment data. Further selection criteria are also necessary and limit the sample to the study of 29 of the 215 state-level minimum wage increases over their time period (their calculation). The donor pools for each of the treated states is limited to states with no minimum wage changes over the same time period and occasionally consist of relatively few states. Furthermore, the synthetic controls are generated for a relatively short time period around the time of adoption of the treated state. These selection criteria are unnecessary using the GSC estimator introduced below as the estimator will include all states and time periods and use all available variation, paralleling additive fixed effects estimators. More importantly, Dube and Zipperer (2015) create the synthetic controls for each adopting state based on pre-intervention outcomes. These outcomes, however, are themselves treated given the potential that the minimum wage causally affects employment. Consequently, the synthetic control for each treated state using CSSC is, in fact, only valid if the minimum wage does not affect employment and, otherwise, does not estimate the model represented in equation (1). The GSC estimator introduced below accounts for the causal impact of the treatment variables when creating the synthetic controls for each state. Overall, it is difficult to use the traditional CSSC estimator to study the effects of the minimum wage. The minimum wage is constantly changing across states, making it difficult to isolate “treated” states and “donor” states. More generally, this problem arises whenever there are multiple treatments, multiple treated units, treatment variables that are continuous, etc. In a fixed effects specification, the designation of units as “treated” and “control” is unnecessary. This paper introduces a synthetic control equivalent.

11

3

Synthetic Control Estimator

This section develops the GSC estimator which generalizes CSSC beyond case studies. It permits the inclusion of multiple treatment variables, which can be discrete or continuous, in the specification. I follow the notation developed for CSSC and used in Section 2.1 as closely as possible. I assume that there are N units and T time periods.3 I model outcomes as Yit = Dit α0 + δt + λt μi + it ,

(5)

where Dit is a K × 1 vector of treatment variables for state i at time period t. As before, λt is a 1 × F vector of common unobserved factors and μi (F × 1) is a vector of factor loadings. The power of the λt μi term parallels the benefits discussed for CSSC. This term permits flexible, non-linear state-level trends and shocks that may be correlated with the policy variables. Similar to the CSSC estimator, the additive fixed effects equivalent to equation (5) is actually a special case and available if that specification is correct. While a state-time interaction term would be collinear with Dit , interactive fixed effects permit the inclusion of a term which varies flexibly at the state and time level. The primary difference between equation (5) and the specification represented in equation (1) is that Dit is a vector and not limited to one indicator variable.4 The estimator will require simultaneous estimation of a synthetic control for each state. Let wi = (wi1 , . . . , wii−1 , wii+1 , . . . , wiN ), where wi is the vector of weights on all other states to generate the synthetic control for 3

It is straightforward to apply the estimator to unbalanced panels. The parameter α0 in equation (5) does not vary by state or time (unlike the parameter in equation (1)). However, it is straightforward to allow this heterogeneity by interacting Dit with state or time indicators. 4

12

state i. The weights are constrained, as before, to be non-negative and to sum to one. I define the set of possible weighting vectors to create the unit i synthetic control by

Wi =

⎧ ⎨ ⎩

wi

⎫ ⎬   j j w = 1, w ≥ 0 for all j .  i i ⎭

(6)

j=i

wij represents the weight given unit j for the creation of the synthetic control for unit i.

3.1

Main Assumptions

In this section, I discuss the assumptions necessary for identification of α0 . The weights themselves are not necessarily identified as multiple sets of weights may be appropriate. This will not affect identification of the parameters of interest. Let Di = (Di1 , . . . , DiT ). A1 Outcomes: Yit = Dit α0 + δt + λt μi + it A2 Conditional Independence: There exists wi ∈ Wi such that the following conditions hold: (a)

(b)

  j w i μj E μi −

⎡ E ⎣it −

| Di −



j=i

j=i





wij jt

j=i

| Di −

j=i

13

 wij Dj = 0

⎤ wij Dj ⎦ = 0

A3 Rank Condition: ⎡ ⎢ ⎢ E⎢ ⎢ ⎣

(Di1

−

(DiT −



j  j=i wi Dj1 )

.. .  j=i



⎤ j j=i wi Yj1

.. . wij DjT )

 j=i

wij YjT

⎥ ⎥ ⎥ is rank k + 1 for all wi ∈ Wi ⎥ ⎦

A1 defines the outcome as a function of the treatment variables, a time fixed effect, (an arbitrary number of) interactive fixed effects, and an observation-specific disturbance term. The advantage of the synthetic control estimator is that it accounts for λt μi , which may be arbitrarily correlated with the treatment variables. As CSSC does for the comparative case study context, this specification nests additive fixed effect models. The dimensions of λt and μi are both unknown and neither is estimated. A2 assumes the existence of synthetic controls but allows the synthetic control to be imperfect (unlike equation (2)). This assumption states that there exists synthetic control  weights such that Di − j=i wij Dj is uncorrelated with deviations from μi . Of course, this condition is more likely to hold when the deviations of μi from the synthetic control are small. A2(a) does not assume uniqueness of the synthetic control for each unit and, in fact, identification of α0 does not require unique synthetic controls. A2(b) states that the  observation-specific disturbance is conditionally independent of Di − j=i wij Dj . A3 is a rank condition. A necessary condition implied by A4 is T ≥ k + 1. A4  requires independent variation in each component of Dit − j=i wij Djt . Furthermore, it  requires independent variation in λt μi , corresponding to the last column ( j=i wij Yjt ). Alternatively, I could write the rank condition as a function of λt μi terms. I use the above

14

condition since it is less restrictive. The alternative rank condition would use ⎤ ⎡ 

j   ⎢ (Di1 − j=i wi Dj1 ) ⎢ .. E⎢ . ⎢ ⎣ 

(DiT −

j  j=i wi DjT )

λ 1 μ1 .. .

. . . λ1 μi−1 .. .. . .

λ1 μi+1 .. .

. . . λ1 μ N .. .. . .

λT μ1 . . . λT μi−1 λT μi+1 . . . λT μN

However, it is not necessary to require independence of Di1 −

 j=i

⎥ ⎥ ⎥ ⎥ ⎦

wij Dj1 to the full set of

interactive fixed effects for each unit. Some linear combinations of these interactive fixed effects are not allowed because they are excluded from Wi . A4 limits the rank condition to only the permitted weighted combinations.

3.2

Estimation

The estimator simultaneously estimates the parameters associated with each treatment variable while constructing a synthetic control for each unit.

The synthetic control is

 an estimate of Yit − Dit α ˆ using a constrained weighted average of (Yjt − Djt α) ˆ for all      j  ˆ acts as an estimate of λt μi , constructed empirically to j = i. j=i wi Yjt − Djt α

minimize the sum of the square of the residuals. Under the assumptions listed above,    j  − D α ˆ does not have to be a perfect estimate of λt μi . w Y jt jt i j=i This approach generates a synthetic control for each unit while accounting for the causal effects of the policy variables. This joint estimation allows for the weighted average    of Yjt − Djt α ˆ to account for λt μi .

3.2.1

Implementation

The estimator chooses an estimate for α0 and simultaneously constructs a synthetic control for each state. This approach is similar in spirit to an additive fixed effect estimator which jointly estimates the parameters of interest while using an unweighted average of all states 15

as a control for each unit. The synthetic controls are constructed by selecting weights in Wi .   Let |·| represent the Euclidian norm. The estimator is ˆ w ˆ 1, . . . , w ˆN ) = (α,

⎫ ⎧ ⎬ ⎨        j     Y φ Y argmin − D b − − D b  it  jt it jt i ⎭ b,φ1 ∈W1 ,...,φN ∈WN ⎩ t i



 j=i

wˆij



Yjt −

 Djt α ˆ

(7)

j=i

 is used as a “control” for unit i. It would be inappropriate to

construct a synthetic control based solely on Yit since Yit is causally impacted by Dit for all units. In practice, I implement the above minimization in steps. I create a grid of possible values for α0 and search over that grid. For each “guess” b, I calculate Yit − Dit b for all i and then estimate the synthetic control for each unit i. The estimation of each synthetic control involves a constrained minimization. Then, I calculate the argument in equation (7). ˆ The b that minimizes this objection function is α. This procedure assumes that there are only one or two treatment variables such that grid-searching is possible. For each b, one can simply use the synth command5 , with Yit − Dit b as the outcome, developed for the CSSC estimator to obtain a synthetic control for each state. This approach estimates the parameters jointly. Other estimation methods of the synthetic control weights and the parameters of interest are also possible. 5

More information can be found at http://web.stanford.edu/ jhain/synthpage.html

16

3.2.2

Weighting

With CSSC, it is customary to visually check the fit of the synthetic control with the treated unit in the pre-period. It may not be possible to generate an appropriate synthetic control for the treated unit, casting doubts in those circumstance on the estimate generated by CSSC. For the estimator of this paper, the parallel concern is that Assumption A2(a) may not hold for all units. It should be noted that additive fixed effect models assume a stricter version of Assumption A2(a). I employ a two-step procedure which places more weight on units with “more appropriate” synthetic controls. The steps are as follows: 1. For each state i, ⎧ ⎫ ⎨     ⎬      i ≡ min bi  φji Yjt − Djt Ω Yit − Dit bi − bi ,φi ∈Wi ⎩ ⎭ t

j=i

In the first step, I allow the parameters (bi ) to vary by state. The purpose of the first step is simply to estimate a measure of fit by state. If a unit or units do not have valid synthetic controls, then these may affect the first-step estimate of α0 and the measure of fit for all states.6 Instead, I estimate the sum of squared errors by state so that the units without valid synthetic controls do not impact the variance estimate for the other units.  −1 . 2. Let Ai ≡ Ω i 3. Minimize the weighted sum of squared residuals:

ˆ w ˆ 1, . . . , w ˆN ) = (α, 6

It is possible that α0 is not identified when using only a single unit. However, this will not cause problems in estimating the variance for each unit.

17

⎧ ⎫ ⎨     ⎬       argmin b  φji Yjt − Djt Yit − Dit b − Ai ⎭ b,φ1 ∈W1 ,...,wN ∈WN ⎩ i

t

(8)

j=i

This method will reduce the weight on units with poor synthetic controls. I employ this two-step method in the simulations.7

3.2.3

Discussion

The primary motivation of GSC is to extend synthetic control estimation to applications with multiple treatment variables, multiple adopters, non-binary treatments, etc. It is instructive to discuss the usefulness of GSC in more basic cases, including those in which CSSC is typically used, such as applications with multiple adopters of the same policy or even with case studies when conditioning on additional covariates is important for identification. It is common to apply the CSSC estimator to applications where multiple units adopt the same treatment. For each treated unit, a separate treatment effect is estimated. However, there is no accepted method for aggregating the multiple estimates. GSC aggregates these estimates naturally by estimating a single parameter across all treated units.8 Furthermore, when multiple units are treated, it is unclear how to construct the donor pools for each treated unit. For example, imagine a policy which all 50 states adopt over the sample period but at different times. The donor pool is empty under the requirement that the donor pool consists only of untreated states. With a fixed effects model, it is unnecessary to separate the states into ever-treated states and never-treated states. With CSSC, however, this distinction is required. Donohue et al. (2015) studies right-to-carry 7

The weighting procedure makes little difference in the final estimates of the minimum wage application (see Table 3). Though not shown, the simulation results (Tables 1 and 2) were also similar with and without the second step of the two-step approach. 8 If unit-specific estimates are desired, they are still available in this framework by interacting the treatment variables with unit-specific indicators.

18

(RTC) laws. Only 9 states had never passed RTC legislation by 2012 and the authors hypothesize that is too few to construct proper synthetic controls for each adopting state and to conduct appropriate inference. To expand the donor pool, Donohue et al. (2015) include some adopting states in the donor pool based on the time of adoption. Similarly, Billmeier and Nannicini (2013) expands the donor pool for their application by restricting the analysis to countries with a sufficient number of countries which were non-treated within 10 years of the treated country (meeting other restrictions as well). The synthetic control estimator introduced above parallels models with additive fixed effects. Treated units are (potentially) also used as controls when appropriate. Dividing units into treatment units and control units is unnecessary. The GSC estimator combines the benefits of CSSC (interactive fixed effects) with the benefits of traditional panel data models (using all states as possible controls). Moreover, a major appeal of CSSC is the ability to construct a control unit with the same pre-treatment pattern of the outcome variable. Abadie and Gardeazabal (2003) and Abadie et al. (2010) model the matching procedure to generate the synthetic control weights as a function of pre-treated outcomes and covariates. However, Kaul et al. (2015) points out that if each pre-treatment outcome is used for the matching, then the covariates will have no effect on the synthetic control and effectively drop out. Kaul et al. (2015) proposes that only one pre-treatment outcome or the average pre-treatment value of the outcome variable should be used. This approach undermines a primary motivation for using CSSC. The introduced framework allows for the outcome to be a function of multiple variables and, consequently, it is possible to create a synthetic control for the treated unit while also accounting for other time-varying covariates. These covariates are not used to create the synthetic control but can have their own independent effects on the outcome. Again, this generalizes traditional additive fixed effect models.

19

3.3

Identification

Identification of α0 holds under the assumptions listed above. These assumptions are similar to those for many panel data estimators with the exception that the treatment variables must have independent variation relative to the synthetic controls (i.e., relative to the weighted average of the other units). In an additive fixed effect model, the corresponding assumption is that the treatment variables must have independent variation relative to the unweighted average of the other units.      j     Theorem 3.1 (Identification). If A1-A3 hold, then E Yit − Dit b − j=i wi Yjt − Djt b  has a unique minimum and b = α0 at this minimum. I include a discussion in Appendix Section A.1. Theorem 3.1 only states that α0 is unique. The weights are not necessarily unique but unique weights do not impact identification of the treatment effects.

3.4

Consistency

Throughout this paper, I consider the weights which generate the synthetic controls as nuisance parameters which are necessary for consistent estimation of α0 . To discuss properties of the GSC estimates, however, it is helpful to make assumptions about the synthetic untreated outcomes, which I denote Γit (b). For a given b, it is possible to estimate the corresponding synthetic control weights for each unit. Define

Γit (b) ≡

N 

  wji (b) Yjt − Dit b ,

j=1

   T N        i   φj Yjt − Djt b  . where wi (b) = argmin Yit − Dit b −  φi ∈Wi t=1   j=1  20

(9)

The wji (b) are defined by equation (9). These are the synthetic control weights if the treatment effects were equal to b. I discuss consistency for T → ∞ and assume that N is fixed. Consistency of α0 requires additional assumptions on the data: A4: (Yit , Dit ) stationary and ergodic for each i.   A5: Define Θ = b | Yit − Dit b ≤ Yit for all i, t and assume Θ is compact with α0 in the interior of Θ. A6: Yit < ∞. A7: Γit (b) continuous in b and twice differentiable. A4 permits dependence within each unit. Other dependence structures would also generate similar theoretical results. I impose no assumptions about independence or dependence across units. The inference procedure will allow for dependence across units and arbitrary dependence within-units. A5 assumes that the parameter space is compact. It makes a further restriction that Θ only include b such that Yit − Dit b ≤ Yit . This parameter set is not empty since it holds for b = 0. This assumption makes the discussion of consistency and asymptotic normality more straightforward. A6 states that the norm of the outcome is finite. A7 disallows large jumps in the value of the synthetic control for small changes in the treatment parameters. This assumption does not rule out large changes in the weights for any synthetic control as b changes. Given that the weights are not necessarily unique, we might expect that small changes in the parameters (b) may generate large changes in the synthetic control weights. However, this does not imply large changes in Γit (b). Given that Yit − Dit b is continuous in b and the synthetic control is trying to fit this term, assuming continuity of Γit (b) is likely unrestrictive. Under these assumptions, uniform convergence of the objective function holds (Theorem 2 in Jennrich (1969)) and, consequently, the estimates are consistent by Theorem 2.1

21

of Newey and McFadden (1994): p

 −→ α0 . Theorem 3.2 (Consistency). If A1-A7 hold, then α See Appendix A.1 for a more detailed discussion.

3.5

Asymptotic Normality

The proposed inference procedure uses the gradient of the objective function and relies on the gradient converging to a normally-distributed random variable. This requires asymptotic normality of the α0 estimates. This section focuses primarily on showing that the estimate √ converges at a T rate or faster. As before, I assume fixed N . Because I make no assumptions about dependence across units, it is only possible to say that the estimate converges at least √ as fast as T . Define the gradient for unit i as T 1 gi (b) ≡ − E T t=1



   ∂Γit (b)  Yit − Dit b − Γit (b) Dit − ∂b

Define T 1 ˆ Hi (b) ≡ T t=1



     (b) (b) ∂Γ ∂Γ ∂ 2 Γit (b)  it it Yit − Dit b − Γit (b) + Dit − . Dit − ∂b∂b ∂b ∂b

The Hessian (H), evaluated at α0 , is ! H≡E

∂Γit (α0 ) Dit − ∂b

"!

I impose additional regularity conditions:

22

∂Γit (α0 ) Dit − ∂b

" ,

A8: H full rank. A9: Σ ≡ limT →∞ Var

#1  N

i

$ gi (α0 ) exists.

   1  ∂ 2 Γit (b)     A10: There exist constants C1 and C2 such that  T t ∂b∂b Yit − Dit b − Γit (b)  ≤          it (b) it (b)  ≤ C2 < ∞, for all i and b ∈ Θ. Dit − ∂Γ∂b C1 < ∞ and  T1 t Dit − ∂Γ∂b   A8 is related to A3 but includes independence of the gradient of Γit (b). A9 (with A4) implies that a central limit theorem holds related to the gradient function. A10 is necessary for uniform convergence of the Hessian matrix. Theorem 3.3 (Asymptotic Normality). If A1-A10 hold, then

√

d

 − α0 ) −→ N (0, H −1 ΣH −1 ) M (α

for some M ≥ T . I include a discussion of this theorem in Appendix A.1. Again, I make no assumptions on independence across units so can only show that the estimates converge at a rate √ √ T or faster, though they potentially converge at rate N T . Estimation of Σ is likely complicated when N is fixed and correlations across units exists. Estimation of H is also likely complicated. These concerns motivate the inference procedure discussed below. The inference procedure will not require estimation of H and will account for dependence across units.

3.6

Inference

I discuss an inference procedure that is valid for fixed N as T → ∞. Furthermore, the procedure permits proper inference even if the clusters are correlated. This feature is important in this context and, more generally, whenever a small number of units are (implicitly or explicitly) used as controls for other units because this approach generates a mechanical correlation across clusters. For the synthetic control estimator of this paper, the estima-

23

tor constructs a synthetic control for each unit through a weighted (constrained) average of   using all other units. Consequently, each unit composes part of the synthetic conα Yjt −Djt

trols for other units, generating correlations across units. This section follows the inference procedure developed in Powell (2015) for panel data estimators more generally. The method simulates the distribution of a test statistic using weights generated by the Rademacher distribution which is equal to 1 with probability 1 . 2

1 2

and -1 with probability

The method uses the panel nature of the data to estimate the relationship between the

gradient functions for each cluster and isolates the independent part of the gradient for each cluster. Once the independent functions are isolated, appropriate inference is possible through simulation. For inference, this paper considers a null hypothesis of the form

H0 : a(b) = 0.

(10)

This framework allows for nonlinear hypotheses and follows the notation used in Newey and West (1987). I assume that null hypothesis only involves b and not the synthetic control weights. The null hypothesis will be tested by imposing the restriction when minimizing the objective function:

˜ w ˜ 1, . . . , w ˜N ) = (α,

⎫ ⎧ ⎨      ⎬     argmin φji Yjt − Djt b  subject to a(b) = 0. Yit − Dit b − ⎭ b,φ1 ∈W1 ,...,φN ∈WN ⎩ i

t

j=i

(11) In words, the null hypothesis is imposed and the other parameters (including the synthetic 24

control weights) are estimated. The inference procedure will use the gradient with respect to α for each unit i with the null hypothesis imposed, which I denote

˜ w ˜ i) = gi (α,

1 T

T  t=1

⎡

⎤⎡

⎤      ⎣Dit − ˜− ˜ ⎦. α w˜ji Djt ⎦ ⎣Yit − Dit α w˜ji Yjt − Djt j=i

(12)

j=i

˜ w ˜ i ) is a K × 1 vector. The inference procedure introduced in Powell (2015), uses only gi (α, a subset of these elements (H ≤ K), defined by the conditions below. I denote the function ˜ i ) and introduce conditions that these ˜ w that will be used in the inference method by si (α, functions must meet. I1 (Identification): There exists H × 1 vector si (·) such that

  ˜ i ) = (q1 , . . . , qH ) E si (α, ˜ w

where

  ˜ i) = 0 E si (α, ˜ w

if

a(b) = 0

qh = 0

if

a(b) = 0

for all h

˜ w ˜ i) Assumption I1 is an identification assumption that is met by at least one element of gi (α, under Theorem 3.1. This condition states that only some elements of the gradient will be ˜ w ˜ i ) will equal 0 even when the null hypothesis is not true and used. Some elements of gi (α, these conditions are not useful for inference. Equation (12) also excludes the gradient of the objective function with respect to the synthetic weights. Under the null hypothesis and Theorem 3.3, we know that

√

d

˜ −→ N (0, V˜i ) T si (α, ˜ w)

for some V˜i .9 Given arbitrary within-unit dependence, estimation of V˜i is difficult. Instead, I perturb the si functions given weights meeting the following conditions. ˜ i ) such that for all i I2 (Weights): Assume i.i.d weights {Wi }N ˜ w i=1 independent of si (α, a. E[Wi ] = 0. 9

This result follows from Theorem 3.3 and continuity of the gradient under A7.

25

b. E[Wi2 ] = 1. c. E |Wi |r < Δ < ∞ for all i. ˜ i ) for each i. Given these weights, it is possible to simulate the distribution of si (α, ˜ w I use the Rademacher distribution which has the advantage that E[Wi3 ] = 0 and E[Wi4 ] = 1 such that using this distribution is able to asymptotically match symmetric distributions (such as the normal distribution) and provide asymptotic refinement.10 Condition I2c is necessary to ensure that a CLT holds for the weighted functions. Finally, I assume that the gradient functions across clusters are asymptotically independent: # $ p ˜ i ), sj (α, ˜ j ) −−−−→ 0 for all i = j as T → ∞. I3 (Asymptotic Independence): Cov si (α, ˜ w ˜ w This assumption is likely not met by the gradient function for GSC. However, it is possible to construct a function that meets this condition using the gradients. Before I discuss that step, I include a result from Powell (2015):11 Lemma 3.1. Assume a(b) = 0 and that A1-A10, I1-I3 hold with T → ∞. Then, ⎛ ˜ ˜ w) ⎜ s1 (α, √ ⎜ .. T⎜ . ⎜ ⎝ ˜ ˜ w) sN (α,

⎞

⎟ ⎜ ⎟ d ⎜ ⎟ −−−−→ N ⎜0, ⎟ ⎜ ⎠ ⎝

⎛ ˜ ˜ w) ⎜ W1 s1 (α, √ ⎜ .. T⎜ . ⎜ ⎝ ˜ WN sN (α, ˜ w)

⎞

⎛ V˜1 ..

.

0 ⎟ ⎟ V˜N

0

⎟ ⎟ ⎠

⎛

⎞

⎜ ⎟ ⎜ ⎟ d ⎟ −−−−→ N ⎜0, ⎜ ⎟ ⎝ ⎠

⎞ V˜1 ...

0

0 ⎟ V˜N

⎟ ⎟ ⎟ ⎠

The importance of Lemma 3.1 is that both vectors converge to the same distribution. ˜ i) Because E[Wi Wj ] = 0, the convergence to the same distribution can only occur if si (α, ˜ w 10

If the number of units is less than 10, Powell (2015) recommends using the weights introduced in Webb (2013). 11 Lemma 2.2 in Powell (2015).

26

˜ j ) are asymptotically uncorrelated since any correlation would not be preserved and sj (α, ˜ w by the weights. For inference, the method uses a Wald Statistic defined by: ⎛ ⎞ ⎞ N N   1 1  α, ˜ −1 ⎝ ˜ i )⎠ Σ( ˜ i )⎠ , ˜ w) si (α, ˜ w si (α, ˜ w S=⎝ N i=1 N i=1 ⎛

 α, ˜ = where Σ( ˜ w)

1 N −1

N i=1

! ˜ i) − ˜ w si (α,



1 N

N j=1

˜j) sj (α, ˜ w

 " !

(13)

˜ i) − ˜ w si (α,



1 N

N j=1

˜j) sj (α, ˜ w

This approach does not require estimation of V˜ , permitting arbitrary within-cluster correla˜ i ) across clusters. tions. Instead, it uses the variance of si (α, ˜ w (d)

To simulate the distribution of the test statistic, consider weights Wi

which satisfy

condition I3 and construct a simulated test statistic, indexed by (d): ⎛ ⎞ ⎞ N N   1 1 (d) (d)  (d) (α, ˜ i )⎠ Σ ˜ −1 ⎝ ˜ i )⎠ , =⎝ Wi si (α, ˜ w ˜ w) W si (α, ˜ w N i=1 N i=1 i ⎛

S (d)

(14)

⎛ ⎛ ⎞⎞ ⎛ ⎛ ⎞⎞ N N N    1 1 1 ⎜ (d) ⎟ ⎜ (d) ⎟ (d) (d)  (d) (α, ˜ = ˜i) − ⎝ ˜ j )⎠⎠ ⎝Wi si (α, ˜i) − ⎝ ˜ j )⎠⎠ . where Σ ˜ w) ˜ w W sj (α, ˜ w ˜ w W sj (α, ˜ w ⎝Wi si (α, N − 1 i=1 N j=1 j N j=1 j

The scores are perturbed such that the test statistic can be simulated without ˜ i ) should be centered around ˜ w re-estimating the model. Under the null hypothesis, si (α, zero such that large values (in magnitude) suggest that the null hypothesis is incorrect. (d)

˜ i ) is centered around zero whether the null hypothesis is true or not. ConseWi si (α, ˜ w quently, when the null hypothesis is not true, the large value of S should be rare in the distribution of S (d) . Theorem 3.4. Assume a(b) = 0 and that A1-A10, I1-I3 hold with T → ∞. Then, S and S (d) converge to the same distribution. Theorem 3.4 follows from Lemma 3.1 and the continuous mapping theorem. See

27

" .

Powell (2015) for more details.

3.6.1

Adjusting for Correlations

The above results assume asymptotic independence across clusters. It is necessary to construct asymptotically independent functions for each unit using the gradient. I model each element of the gradient (denoted by k) for unit i as correlated with the gradient for all other units by

˜ i) = gitk (α, w

L 

 aik Mtk ,

 with E[Mtk ] = 0 for all ,

˜ ˜ (15) E[Mt Mtk ] = 0 for  = .

=1

Given that no restrictions are placed on aik , this assumption is not restrictive in terms of constraining the variance of git . Furthermore, no restrictions are placed on the relationship between Mt and Ms for all s and t, permitting arbitrary within-cluster correlations. Instead, this assumption rules out that lags or leads of the M random variables independently affect a unit without proportionately affecting other units. This condition is met if it and js are uncorrelated for i = j or t = s. It is straightforward to extend this approach to allow for such leads and lag, though this extension is not pursued here.12 This setup permits correlations driven by the synthetic control approach. Furthermore, it allows for arbitrary within-time correlations across units which are determined by outside (non-estimation) factors such as common shocks. States may have correlated gradients because they experience similar shocks and trends due to geographic proximity, similar political institutions, comparable industry composition, etc. The inference approach will account for correlations regardless of the underlying mechanisms. I rewrite the above setup in a triangular framework such that gitk is only a function of 12

See Powell (2015) for more about this extension.

28

gi+1,tk , . . . , gN,tk . This is a simple rearrangement of equation (15) and imposes no additional assumptions. ˜ i) = gitk (α, ˜ w

N 

˜ i ) + μitk , cijk gjtk (α, ˜ w

(16)

j=i+1

where μitk is uncorrelated with μjtk for all i = j. The above setup separates the gradi i ˜ i )) and an independent ent of each unit into a dependent component ( N ˜ w j=i+1 cjk gjtk (α, component (μitk ). Using OLS, estimate cˆijk in equation (16) for all i, j, k. Then, ⎛ Cov ⎝gitk −

N  j=i+1

cˆijk gjtk ,

N 

gmtk −

⎞ ⎠ −−−−→ 0 for all i = m. cˆm jk gjtk p

(17)

j=m+1

Estimation of equation (16) allows for the correlations across clusters to be estimated. It is then possible to create independent functions for each cluster. Given asymptotically uncorrelated functions, Theorem 3.4 holds. Alternatively, it is also possible to estimate ⎛ ˜ i ) = cik ⎝ gitk (α, ˜ w

N 

⎞ gjtk ⎠ + μitk .

(18)

j=i+1

The cijk terms are not parameters of interest so it is possible to aggregate them into one estimate. The inference steps are as follows: 1. Estimate the constrained parameters using equation (11). 2. Create the gradient for each unit using equation (12) and meeting assumption I1. 3. Estimate equation (16) (or equation (18)) for each unit and form the residual. These are the si functions. 4. Create the test statistic defined by equation (13). 5. Simulate the test statistic using Rademacher weights 999 times (equation (14)).

29

6. The p-value is equal to the number of times that S is greater than the simulated value  (d) ). of S: pˆ = D1 D d=1 1(S > S Large values of S imply that we should rarely observe the distribution of gradients created by the null hypothesis and that we should reject it. By Theorem 3.4, we can simulate how often we should see a value as large as S if the null hypothesis were true.

4

Simulations

In this section, I report the results of simulations using the synthetic control estimator of this paper and the more traditional additive fixed effects estimator. I generate data with two sets of interactive fixed effects. First, I generate data with a continuous treatment effect which is a function of the interactive fixed effects. Second, I create a standard difference-in-differences situation where the treatment is represented by a dummy variable.

4.1

Continuous Treatment Effect

The data are generated by (1) (1)

(2) (2)

Treatment Variable: dit = φit + μi λt + 2μi λt (1) (1)

(2) (2)

Outcome: yit = 2μi λt + μi λt + it

(1)

(2)

where μi , μi

(1)

∼ U (0, 1); λt

(2)

∼ N (0, 400); λt

∼ N (0, 100); φit ∼ U (0, 50); it ∼ N (0, 1).

Both the outcome and the treatment variable are functions of the interactive fixed effects. Not accounting for these interactive fixed effects should lead to biased estimates. I generate

30

the above data for N = 30 and T = 30. The treatment effect is zero, and the treatment variable is continuous. In the top part of Table 1, I report Mean Bias, Median Absolute Deviation, and Root-Mean-Square Error (RMSE). The first estimator is an additive fixed effects estimator which includes both state and time fixed effects. The second estimator is the synthetic control estimator of this paper. The fixed effects estimator, as expected, performs poorly. The synthetic control estimator performs well. In the bottom half of Table 1, I test the inference procedure discussed in Section 3.6. For the additive fixed effects estimator, I adjust the standard errors for clustering at the state level and use a t-distribution with 29 degrees of freedom. The fixed effects estimator always rejects the null hypothesis (α0 = 0). The inference procedure of Section 3.6 using the synthetic control estimates works well at multiple significance levels. Table 1: Simulation Results: Continuous Treatment Variable

Fixed Effects Synthetic Control

Fixed Effects Synthetic Control

Mean Bias

Median Absolute Deviation

Root Mean Squared Error

0.2988 0.0011 Rejection Rate Significance Level = 0.10 1.000 0.099

0.2966 0.0000 Rejection Rate Significance Level = 0.05 1.000 0.055

0.3055 0.0061 Rejection Rate Significance Level = 0.01 1.000 0.009

The “fixed effects” estimator refers to a model with additive state and time fixed effects. For inference, standard errors are adjusted for clustering at the state level and a t-distribution with 29 degrees of freedom is used. The “synthetic control” estimator is introduced in this paper. The inference procedure is discussed in Section 3.6.

4.2

Difference-in-Differences

I generate similar data as above but use a discrete treatment variable. This treatment variable is equal to 0 for all units initially. Once the treatment is adopted, the unit never repeals the treatment. Finally, all units adopt treatment by period t = 28. It would be 31

difficult to use CSSC to estimate the relevant parameter with these data since it is not clear which states to consider “treated” and which to include as part of the donor pool for those treated units. The data are generated by   (1) (1) (2) (2) Define ait = 1 φit + μi λt + 2μi λt > 45 ⎧ ⎪ ⎪ ⎪ 0 if t = 1 ⎪ ⎪ ⎪ ⎨ Treatment Variable: dit = 1 if di,t−1 = 1 or ait = 1 or t ≥ 28 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 otherwise (1) (1)

(2) (2)

Outcome: yit = 2μi λt + μi λt + it

(1)

(2)

where μi , μi

(1)

∼ U (0, 1); λt

(2)

∼ N (0, 400); λt

∼ N (0, 100); φit ∼ U (0, 50); it ∼ N (0, 1).

This data generating process produces a serially-correlated treatment dummy variable equal to 1 about 50% of the time. The simulation results are presented in Table 2. The additive fixed effects estimator performs especially poorly as one would expect given the correlation between the (omitted) interactive fixed effects and the treatment variable. The synthetic control estimator performs well. It also rejects at approximately the correct rates.

5

Employment Effects of Minimum Wage

Recent work has suggested that the traditional empirical strategy in the minimum wage literature – using all states as controls for states with increasing minimum wages – is inappropriate. The synthetic control estimator of this paper will empirically generate an appropriate control for each state. I first replicate findings using the additive fixed effect

32

Table 2: Simulation Results: Difference-in-Differences

Fixed Effects Synthetic Control

Fixed Effects Synthetic Control

Mean Bias

Median Absolute Deviation

Root Mean Squared Error

2.0639 -0.0013 Rejection Rate Significance Level = 0.10 0.628 0.089

2.6580 0.0200 Rejection Rate Significance Level = 0.05 0.567 0.052

4.2160 0.0277 Rejection Rate Significance Level = 0.01 0.386 0.005

The “fixed effects” estimator refers to a model with additive state and time fixed effects. For inference, standard errors are adjusted for clustering at the state level and a t-distribution with 29 degrees of freedom is used. The “synthetic control” estimator is introduced in this paper. The inference procedure is discussed in Section 3.6.

specification typically estimated in the literature:

ln Est = αs + γt + β ln(M Wst ) + st

where Est is the employment rate in state s, quarter t for 16-19 year olds. The log of the employment rate is modeled as a function of state fixed effects, time fixed effects, and the log of the minimum wage (M W ). I do not control for the overall state unemployment rate or the relative size of the youth population, though these controls are typical in the literature. These controls are potentially problematic if they are also affected by changes in the minimum wage and are often included because of concerns that minimum wage policy reacts to state economic conditions. An advantage of the synthetic control estimator is that it should make inclusion of these control variables unnecessary. I use the data set constructed for Dube and Zipperer (2015) and Allegretto et al. (2015).13 This allows for straightforward comparisons of estimates generated by the synthetic control approach and fixed effects estimation. The data are state-level minimum wage 13

Code and data can be found at Arindrajit Dube’s site: http://arindube.com/working-papers/, accessed January 15, 2016.

33

information merged with employment rates aggregated from the Current Population Survey (CPS) for 1979-2014. The data are quarterly and aggregated by state. The outcome of interest is the employment rate of the 16-19 population. There are 51 units and 144 time periods. Table 3 replicates results found in Table 2 of Allegretto et al. (2015) without conditioning on the control variables mentioned above. Columns (1)-(6) show that the estimate varies between -0.243 and 0.083 depending on the inclusion and flexibility of state-specific trends. Columns (7)-(12) include division-time fixed effects such that identification originates from minimum wage changes within a Census division. The estimates vary between -0.228 and 0.125. In general, the estimates appear sensitive to the inclusion of state-specific trends and choice of control group. GSC should account for these considerations. I report the main empirical results of this paper in Columns (13) and (14) of Table 3. In column (13), I report GSC estimates which weight all states equally, regardless of the fit of the synthetic control generated for each state. I estimate an elasticity of -0.45, larger in magnitude than estimates typically found in the literature. In Section 3.2.2, I discussed the merits of a two-step procedure which places more weight on states with better synthetic controls. Such a two-step procedure could also be employed for fixed effect estimators but this is not typically done in applied work. Column (14) reports the estimates using the two-step procedure. I estimate an elasticity of -0.44, implying that a 10% increase in the minimum wage decreases the teen employment rate by 4.4%. This estimate is statistically significant from zero at the 1% level. These estimates are larger in magnitude than those typically generated by additive fixed effect models, suggesting that those models produce estimates that are biased against finding a relationship between the minimum wage and employment.

34

Table 3: Effect of Minimum Wage on Teen Employment Outcome:

ln(Minimum Wage) State Fixed Effects Time Fixed Effects State Trends Division-specific Time Effects N

ln(Minimum Wage) State Fixed Effects Time Fixed Effects State Trends Division-specific Time Effects N

ln(Minimum Wage) Two-Step Weighting? N

ln(Employment / Population) Additive State and Time Fixed Effects (2) (3) (4) (5) 0.044 0.053 0.083 0.017 [-0.120, 0.208] [-0.139, 0.245] [-0.097, 0.262] [-0.174, 0.209] Yes Yes Yes Yes Yes Yes Yes Yes Linear Quadratic Cubic Quartic No No No No 7344 7344 7344 7344 Additive State and Division-Time Fixed Effects (7) (8) (9) (10) (11) -0.228* 0.106 0.125 0.105 0.077 [-0.469, 0.013] [-0.056, 0.267] [-0.105, 0.356] [-0.070, 0.281] [-0.099, 0.252] Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Linear Quadratic Cubic Quartic Yes Yes Yes Yes Yes 7344 7344 7344 7344 7344 Synthetic Control Estimates (13) (14) -0.45** -0.44*** [-1.19, -0.02] [-0.77, -0.02] No Yes 7344 7344

(1) -0.243*** [-0.406, -0.080] Yes Yes No No 7344

(6) 0.026 [-0.166, 0.218] Yes Yes Quintic No 7344 (12) 0.084 [-0.098, 0.266] Yes Yes Quintic Yes 7344

Notes: Significance Levels: *10%, **5%, ***1%. 95% Confidence Intervals in brackets are adjusted for clustering at statelevel. Hypothesis testing for synthetic control estimator use simulation technique discussed in text. Inference for the synthetic control estimates accounts for within-state clustering and cross-sectional correlations. 2-step weighting uses the estimation procedure discussed in Section 3.2.2.

35

6

Discussion and Conclusion

This paper introduces a synthetic control estimator which generalizes the case study synthetic control estimator of Abadie and Gardeazabal (2003) and Abadie et al. (2010). It also generalizes traditional additive fixed effect models. The estimator jointly estimates the parameters associated with the treatment variables while creating synthetic controls for each unit. These synthetic controls permit the treatment variables to co-vary with flexible state-specific trends. The estimator works well in simulations. I provide evidence of the usefulness of the estimator by estimating the effect of the minimum wage on teenage employment rates. The recent literature on this topic has noted that choice of appropriate control groups for states increasing their minimum wages is essential for consistent estimation. The estimator of this paper creates these control groups empirically instead of assuming that all other states are appropriate as additive fixed effect estimator assume. The estimator in this paper should be useful more broadly for applications using panel data.

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A

Appendix

A.1

Properties of Estimator

     j     Theorem 3.1 (Identification). If A1-A3 hold, then E Yit − Dit b − j=i wi Yjt − Djt b  has a unique minimum and b = α0 at this minimum. Proof.

⎡

⎤          E ⎣Yit − Dit b − b ⎦ wij Yjt − Djt j=i

is minimized for b, wi ∈ Wi such that

Dit b

+



 j=i

wij

conditional expectation of Yit . I first show that Dit α0 +





 Djt b



is equal to the Yjt −    j  is the j=i wi Yjt − Djt α0

expected value of Yit . ⎡ E ⎣Yit − Dit α0 − ⎡

⎛

⎞



wij



⎤   j  w i Dj ⎦ Yjt − Djt α0 Di −

j=i

j=i

⎤

 j  j   j ⎥ ⎢ wi μj ⎠ + it − wi jt Di − w i Dj ⎦ = E ⎣λt ⎝μi − j=i

j=i

by A1

j=i

= 0 by A2.

The above implies that ⎡

⎤ ⎤ ⎡         E ⎣Yit Di − wij Dj ⎦ = E ⎣Dit α0 + wij Yjt − Djt α0 Di − wij Dj ⎦ j=i

⎡⎛

j=i

j=i

⎤   j  j   j ⎥ ⎢ = E ⎣⎝Dit − wi Djt ⎠ α0 + wi Yjt Di − w i Dj ⎦ j=i

37

⎞

j=i

j=i

By assumption A3, we know that this is unique. p

 −→ α0 . Theorem 3.2 (Consistency). If A1-A7 hold, then α Proof. By the triangle inequality,           j          wi Yjt − Djt wij Yjt − Djt b  ≤ Yit − Dit b +  b  Yit − Dit b − j=i

j=i

        j ≤ Yit  +  wi Yjt  By A5 j=i

        ≤ Yit  +  max Yjt  By definition of Wi j

< ∞ By A6

This shows that we can find a function d(Y ), with E[d(Y )] < ∞, which dominates      b for all b. Given that the parameter set is compact by Yit − Dit b − j=i wij Yjt − Djt      b is continassumption, identification of α0 holds, and Yit − Dit b − j=i wij Yjt − Djt uous (by A7), then uniform convergence of the objective function to its expectation holds (Theorem 2 in Jennrich (1969)). ˆ follows immediately from Theorem 2.1 of Newey and McFadden Consistency of α (1994).

Theorem 3.3 (Asymptotic Normality). If A1-A10 hold, then for some M ≥ T . 1. Consistency holds by Theorem 3.2. 2. α0 is in the interior of Θ by A5. 38

√

d

 − α0 ) −→ N (0, H −1 ΣH −1 ) M (α

3. The objective function is twice differentiable by A7. ,  d 4. By condition A9, M i gi (b) −→ N (0, Σ). N ˆ it (b) for all b and, consequently, Finally, by A10, a dominance condition holds for H ˆ i (b) uniformly converges to H(b) by Theorem 2 in Jennrich (1969). By A8, H(α0 ) is H nonsingular. Under these conditions, Theorem 3.1 of Newey and McFadden (1994) holds, proving the theorem.

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